Limiting Absorption Principle for Schr\"odinger Operators with Oscillating Potential

TL;DR

This paper establishes the limiting absorption principle for Schrödinger operators with complex oscillating potentials, including long-range components, using weighted Mourre theory, and also investigates the absence of positive eigenvalues.

Contribution

It extends the limiting absorption principle to a broader class of oscillating potentials with long-range parts, surpassing previous methods.

Findings

Proves limiting absorption principle for new class of oscillating potentials.

Handles potentials with both long-range and short-range components.

Shows absence of positive eigenvalues in certain cases.

Abstract

Making use of the weighted Mourre theory developed in [GJ1], we show the limiting absorption principle for Schr{\"o}dinger operators with perturbed oscillating potential on appropriate energy intervals. We focus on a certain class of oscillating potentials (larger than the one in [GJ2]) that was already studied in [BD, MU, ReT1, ReT2]. We allow long-range and short-range components and local singularities in the perturbation. We improve known results, the main novelty being the presence of a long-range perturbation. A subclass of the considered potentials actually cannot be treated by the usual Mourre commutator method. Inspired by [FH], we also show, in some cases, the absence of positive eigenvalues for our Schr{\"o}dinger operators.